NEW YORK — One after another, major U.S. corporations have updated anti-discrimination policies to protect gay, lesbian and transgender workers, drawing plaudits from gay-rights groups. There’s one prominent exception: Exxon Mobil Corp.

In the latest rankings of such corporate policies by the Human Rights Campaign, a national gay-rights group, many of Exxon’s Fortune 500 counterparts got scores of 80 or higher on a scale of 100. Exxon, the nation’s largest oil and gas company, became the first firm to get a score below zero.

On Wednesday, in the latest attempt to pressure the company into change, a gay-rights group called Freedom to Work teamed with a high-powered Washington law firm to file what they described as a groundbreaking discrimination complaint against Exxon in Illinois.

The complaint, filed with the state’s Department of Human Rights, says Exxon was sent two nearly identical resumes for a job opening at its office in Patoka, Ill. The only substantive differences were that one of the fictional applicants was clearly depicted as a gay-rights activist, and had higher college and high-school grades than the other applicant.

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According to the complaint, Exxon’s human resources office at its home base in Texas confirmed receipt of both applications, then made several efforts to contact the applicant with the lower grades to set up an interview. The applicant who indicated she was gay received no such follow-ups.

“Exxon has repeatedly claimed they do not discriminate,” said Freedom to Work’s president, Tico Almeida. “We are bringing forward proof they’ve broken the law, and we’re hopeful this compelling case will move them over the tipping point.”

An Exxon Mobil spokesman, Charles Engelmann, said the company was reviewing the complaint and had no immediate comment on its allegations.

“Exxon Mobil’s global policies and processes prohibit all forms of discrimination, including those based on sexual orientation and gender identity, in any company workplace, anywhere in the world,” Engelmann said in an email. “In fact, our policies go well beyond the law and prohibit any form of discrimination.”

The Washington-based law firm Cohen Milstein Sellers & Toll, which often works with advocacy groups on workplace discrimination cases, has provided two of its attorneys to handle the Freedom to Work complaint. One of them, Peter Romer-Friedman, contended that Exxon’s handling of the two job applications was a clear violation of Illinois’ anti-discrimination law.

Illinois is one of 21 states that prohibit workplace discrimination based on sexual orientation. Texas has no such law, and Congress has not passed a nationwide ban.

Romer-Friedman said the tactic of submitting fictional, nearly identical applications is a time-tested, legally accepted technique that has been successful in the past in uncovering bias against blacks and women in hiring and fair-housing cases. The Exxon case marks the first time the tactic has been used to allege anti-gay bias, he said.

The next step, Romer-Friedman said, would be for the Illinois Human Rights Department to investigate the complaint — a process which could take a year. At that stage, Freedom to Work could seek a hearing before the state’s Human Rights Commission or take the case to court.

However, the attorney said it’s likely that Illinois human rights officials would invite Freedom to Work and Exxon to negotiate a settlement before the investigation ends. For Freedom to Work, the goal is to get Exxon to include sexual orientation and gender identity as protected categories in its equal employment opportunity policy, and to implement training programs on how to prevent any such bias.

“Our goal is to settle as quickly as possible, so Exxon will let everyone know they’re going to get a fair shot, no matter who they are or who they love,” Almeida said.

Exxon has been criticized over workplace policies in an era where the vast majority of other large U.S. corporations, including rival oil companies, have adopted policies that earn high scores from the Human Rights Campaign. Most, for example, voluntarily offer health benefits to employees’ same-sex partners; Exxon does not do so.

“We’ve seen public opinion on every issue move toward equality, including in corporate America,” said HRC spokesman Paul Guequierre. “To see one company stick out like this is surprising and frustrating.”

Exxon insists its policies toward gay and lesbian employees are fair and supportive, and rejects the HRC scorecard.

For the past several years, New York Comptroller Thomas DiNapoli — who oversees a state employee retirement fund — has filed a resolution at Exxon Mobil’s shareholder meeting proposing that the company explicitly ban discrimination against gay and transgender workers as part of its equal employment opportunity policy. The same proposal is on the agenda for this year’s meeting on May 29 in Dallas.

The previous resolutions have been defeated at the urging of Exxon management, and the board of directors is again calling for a “No” vote this year.

“Exxon Mobil’s policies go beyond the law and prohibit any form of discrimination,” the proxy statement says. “The board believes the proposal is unnecessary.”

DiNapoli, trustee of a fund with millions of Exxon shares, disputes the company’s assertions. He notes that after Exxon acquired Mobil in 1999, it ended Mobil’s policy of providing health benefits to same-sex partners except those previously covered.

“Each year that this comes up is another year that Exxon has to endure negative press,” said Eric Sumberg, a spokesman for DiNapoli. “It’s never clear why they won’t take that next step.”

After last year’s shareholder meeting, leaders at Resource Center Dallas, the main gay community center in Exxon’s home region, wrote to the company offering their expertise in developing more gay-friendly policies.

Cece Cox, the center’s CEO, said a meeting resulted in June 2012 with several senior Exxon executives, including vice presidents for human resources and for investor relations. Attending with Cox were members of the resource center’s board who had helped develop anti-discrimination policies at the Fortune 500 companies they worked for.

Cox said she came away from the meeting with optimism, but never heard back in any substantive way. “We felt like they were looking at making changes — and then nothing,” she said. “We are really stumped.”

Since 2006, Exxon’s chairman and CEO has been Rex W. Tillerson. Pressure on the company from gay-rights groups has increased during his leadership. Tillerson also is a member of the national executive board of the Boy Scouts of America, and served as its president in 2010-2012, a period in which it resisted pressure to ease its ban on openly gay youth members and adult leaders.

For now, Exxon faces no major shareholder rebellion — votes for DiNapoli’s resolution at last year’s annual meeting accounted for only 20.5 percent of all shares.

Phil Weiss, an energy industry securities analyst with New York-based Argus Research, said the controversies over gay-related employment policies were a minor concern, at most, for most Exxon investors.

However, Amy Myers Jaffe, an energy industry expert at the University of California, Davis, said Exxon might be mistaken to think its current stance won’t be harmful in the long term.

“The millennial generation is very into social equality,” she said. “Exxon is not going to get the top candidates they want, coming out of engineering schools, if their values don’t conform with those of the millennials.”